rastafarral6

2021-11-13

Hi i know this is a really really simple question but it has me confused.
I want to calculate the cross product of two vectors
$\stackrel{\to }{a}×\stackrel{\to }{r}$
The vectors are given by

The vector $\stackrel{\to }{r}$ is the radius vector in cartesian coordinares.
I want to calculate the cross product in cylindrical coordinates, so I need to write $\stackrel{\to }{r}$ in this coordinate system.
The cross product in cartesian coordinates is
$\stackrel{\to }{a}×\stackrel{\to }{r}=-ay\stackrel{\to }{x}+ax\stackrel{\to }{y}$,
however how can we do this in cylindrical coordinates?

Donald Proulx

The radius vector $\stackrel{\to }{r}$ in cylindrical coordinates is $\stackrel{\to }{r}=p\stackrel{\to }{p}+z\stackrel{\to }{z}$. Calculating the cross-product is then just a matter of vector algebra:
$\stackrel{\to }{a}×\stackrel{\to }{r}=a\stackrel{\to }{z}×\left(p\stackrel{\to }{p}+z\stackrel{\to }{z}\right)$
$=a\left(p\left(\stackrel{\to }{z}×\stackrel{\to }{p}\right)+z\left(\stackrel{\to }{z}×\stackrel{\to }{z}\right)\right)$
$=ap\left(\stackrel{\to }{z}×\stackrel{\to }{p}\right)$
$=ap\stackrel{\to }{\varphi }$,
where in the last line we've used the orthonormality of the triad $\left\{\stackrel{\to }{p},\stackrel{\to }{\varphi },\stackrel{\to }{z}\right\}$

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